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Maximizing the practical achievability of quantum annealing attacks on factorization-based cryptography

Olgierd Żołnierczyk

TL;DR

The paper investigates practical cryptanalysis of factorization-based cryptography using quantum annealing in a hybrid classical-quantum framework. It extends the General Number Field Sieve (GNFS), whose cost is $O( ext{exp}((1.923+o(1))(\ln N)^{1/3}(\ln\ln N)^{2/3}))$, by pairing with quantum annealing to locate $B$-smooth relations. It reports the experimental demonstration of a 29-bit factorization on a D-Wave device and provides detailed resource usage, illustrating how hybrid approaches can yield practical speedups over purely classical methods in the near term, while still operating within a subexponential overall complexity. These results inform the realistic timeline for quantum-assisted attacks on cryptography and highlight the need to consider hybrid quantum techniques in security planning.

Abstract

This work focuses on quantum methods for cryptanalysis of schemes based on the integer factorization problem and the discrete logarithm problem. We demonstrate how to practically solve the largest instances of the factorization problem by improving an approach that combines quantum and classical computations, assuming the use of the best publicly available special-class quantum computer: the quantum annealer. We achieve new computational experiment results by solving the largest instance of the factorization problem ever announced as solved using quantum annealing, with a size of 29 bits. The core idea of the improved approach is to leverage known sub-exponential classical method to break the problem down into many smaller computations and perform the most critical ones on a quantum computer. This approach does not reduce the complexity class, but it assesses the pragmatic capabilities of an attacker. It also marks a step forward in the development of hybrid methods, which in practice may surpass classical methods in terms of efficiency sooner than purely quantum computations will.

Maximizing the practical achievability of quantum annealing attacks on factorization-based cryptography

TL;DR

The paper investigates practical cryptanalysis of factorization-based cryptography using quantum annealing in a hybrid classical-quantum framework. It extends the General Number Field Sieve (GNFS), whose cost is , by pairing with quantum annealing to locate -smooth relations. It reports the experimental demonstration of a 29-bit factorization on a D-Wave device and provides detailed resource usage, illustrating how hybrid approaches can yield practical speedups over purely classical methods in the near term, while still operating within a subexponential overall complexity. These results inform the realistic timeline for quantum-assisted attacks on cryptography and highlight the need to consider hybrid quantum techniques in security planning.

Abstract

This work focuses on quantum methods for cryptanalysis of schemes based on the integer factorization problem and the discrete logarithm problem. We demonstrate how to practically solve the largest instances of the factorization problem by improving an approach that combines quantum and classical computations, assuming the use of the best publicly available special-class quantum computer: the quantum annealer. We achieve new computational experiment results by solving the largest instance of the factorization problem ever announced as solved using quantum annealing, with a size of 29 bits. The core idea of the improved approach is to leverage known sub-exponential classical method to break the problem down into many smaller computations and perform the most critical ones on a quantum computer. This approach does not reduce the complexity class, but it assesses the pragmatic capabilities of an attacker. It also marks a step forward in the development of hybrid methods, which in practice may surpass classical methods in terms of efficiency sooner than purely quantum computations will.
Paper Structure (21 sections, 3 theorems, 17 equations, 5 tables)

This paper contains 21 sections, 3 theorems, 17 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Paper organization
  3. Theory
  4. Classical factorization method
  5. Essence of GNFS
  6. Linear algebra stage
  7. Factorization of algebraic elements
  8. Closing notes
  9. Quantum annealing and the QUBO problem
  10. Quantum annealing methods for factorization and determining B-smooth Numbers
  11. Direct factorization
  12. Degree reduction of a polynomial
  13. Detecting smoothness
  14. Multiplication table procedure
  15. Experiment
  16. ...and 6 more sections

Key Result

theorem thmcountertheorem

The factorization into prime ideals of the ideal generated by $(a + b\rho)$, assuming that $\gcd(a,b) = 1$, contains only ideals generated by the pair: $(s, \rho -r)$, where:

Theorems & Definitions (5)

  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • theorem thmcountertheorem